A109298 -id:A109298 - cp's OEIS frontend

A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 625, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2500, 2592, 2916, 3125, 3456, 3888, 4096, 5000, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10000, 10368, 11664, 12500, 13824, 15552

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears more than k times. Such partitions are counted by A115584.

			The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  625: {3,3,3,3}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..2847 (terms up to 10^12)

Cf. A000720, A056239, A062457, A109298, A112798, A115584, A118914, A276078.

Cf. A324524, A324525, A324571, A325128, A325130.

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>PrimePi[p]]&]
With[{k = 4}, m = Prime[k]^(k + 1); s = {}; Do[p = Prime[i]; AppendTo[s, Join[{1}, p^Range[i + 1, Floor[Log[p, m]]]]], {i, 1, k}]; Union @ Select[Times @@@ Tuples[s], # <= m &]] (* Amiram Eldar, Oct 24 2020 *)

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^k * (prime(k)-1)) = 1.58661114052385082598.... - Amiram Eldar, Oct 24 2020

A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

Original entry on oeis.org

1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882

Offset: 1

Gus Wiseman, Feb 10 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    9: {2,2}
   54: {1,2,2,2}
  100: {1,1,3,3}
  120: {1,1,1,2,3}
  125: {3,3,3}
  135: {2,2,2,3}
  168: {1,1,1,2,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.

For mean instead of median we have A359903, counted by A360068.
For distinct indices instead of indices we have A360453, counted by A360455.
For distinct indices instead of multiplicities: A360249, counted by A360245.
These partitions are counted by A360456.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranked by A359889.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median.
A359894 counts partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Median[prix[#]]==Median[Length/@Split[prix[#]]]&]

A324587 Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

Original entry on oeis.org

1, 2, 7, 14, 23, 46, 53, 97, 106, 151, 161, 194, 227, 302, 311, 322, 371, 419, 454, 541, 622, 661, 679, 742, 827, 838, 1009, 1057, 1082, 1193, 1219, 1322, 1358, 1427, 1589, 1619, 1654, 1879, 2018, 2114, 2143, 2177, 2231, 2386, 2437, 2438, 2741, 2854, 2933

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also products of distinct elements of A011757.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    7: {4}
   14: {1,4}
   23: {9}
   46: {1,9}
   53: {16}
   97: {25}
  106: {1,16}
  151: {36}
  161: {4,9}
  194: {1,25}
  227: {49}
  302: {1,36}
  311: {64}
  322: {1,4,9}
  371: {4,16}
  419: {81}
  454: {1,49}
  541: {100}

Cf. A001156, A005117, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A276078.

Cf. A109298, A324524, A324525, A324571, A324572, A324588.

Select[Range[1000],And@@Cases[FactorInteger[#],{p_,k_}:>k==1&&IntegerQ[Sqrt[PrimePi[p]]]]&]

A325761 Heinz numbers of integer partitions whose length is itself a part.

Original entry on oeis.org

1, 2, 6, 9, 15, 20, 21, 30, 33, 39, 45, 50, 51, 56, 57, 69, 70, 75, 84, 87, 93, 105, 110, 111, 123, 125, 126, 129, 130, 140, 141, 159, 165, 170, 175, 176, 177, 183, 189, 190, 195, 196, 201, 210, 213, 219, 230, 237, 245, 249, 255, 264, 267, 275, 285, 290, 291

Offset: 1

Gus Wiseman, May 18 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A002865.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   15: {2,3}
   20: {1,1,3}
   21: {2,4}
   30: {1,2,3}
   33: {2,5}
   39: {2,6}
   45: {2,2,3}
   50: {1,3,3}
   51: {2,7}
   56: {1,1,1,4}
   57: {2,8}
   69: {2,9}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   87: {2,10}

Cf. A001222, A002865, A056239, A093641, A109298, A110295, A112798, A118914, A325762, A325763.

Select[Range[100],MemberQ[PrimePi/@First/@FactorInteger[#],PrimeOmega[#]]&]

A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164, 167, 171, 172, 173

Offset: 1

Gus Wiseman, May 04 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities, starting from the largest part through to the smallest, form an initial interval of positive integers. The enumeration of these partitions by sum is given by A179269.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}

Cf. A055932, A056239, A098859, A109298, A112798, A130091, A179269, A317090, A325326, A325337, A325460.

Select[Range[100],Last/@If[#==1,{},FactorInteger[#]]==Range[PrimeNu[#],1,-1]&]

A276936 Numbers m with at least one distinct prime factor prime(k) such that prime(k)^k divides, but prime(k)^(k+1) does not divide m.

Original entry on oeis.org

2, 6, 9, 10, 14, 18, 22, 26, 30, 34, 36, 38, 42, 45, 46, 50, 54, 58, 62, 63, 66, 70, 72, 74, 78, 82, 86, 90, 94, 98, 99, 102, 106, 110, 114, 117, 118, 122, 125, 126, 130, 134, 138, 142, 144, 146, 150, 153, 154, 158, 162, 166, 170, 171, 174, 178, 180, 182, 186, 190, 194, 198, 202, 206, 207, 210, 214, 218, 222, 225

Offset: 1

Antti Karttunen, Sep 24 2016

nonn

Numbers m with at least one prime factor such that the exponent of its highest power in m is equal to the index of that prime.

The asymptotic density of this sequence is 1 - Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.31025035294364447031... - Amiram Eldar, Jan 09 2021

			2 is a member as 2 = prime(1) and as 2^1 divides but 2^2 does not divide 2.
3 is NOT a member as 3 = prime(2) but 3^2 does not divide 3.
4 is NOT a member as 2^2 divides 4.
6 is a member as 2 = prime(1) and 2^1 is a divisor of 6, but 2^2 is not.
9 is a member as 3 = prime(2) and 3^2 divides 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5500 from Antti Karttunen)

Cf. A109298, A276935.
Intersection with A276078 gives A276937.
Cf. A016825, A051063 (subsequences).
Complement of A325130.

q:= n-> ormap(i-> numtheory[pi](i[1])=i[2], ifactors(n)[2]):
select(q, [$1..225])[];  # Alois P. Heinz, Nov 18 2024

Select[Range[225], AnyTrue[FactorInteger[#], PrimePi[First[#1]] == Last[#1] &] &] (* Amiram Eldar, Jan 09 2021 *)

A325762 Heinz numbers of integer partitions with no part greater than the number of ones.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 432, 448, 480, 512, 560, 576, 600, 640, 648, 672, 704, 720, 768, 784, 800, 832, 864, 896, 960, 972, 1000

Offset: 1

Gus Wiseman, May 18 2019

nonn

After 1 and 2, first differs from A322136 in having 200.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A002865.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    24: {1,1,1,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   128: {1,1,1,1,1,1,1}
   144: {1,1,1,1,2,2}

Cf. A001222, A002865, A007814, A056239, A061395, A093641, A109298, A110295, A112798, A118914, A325761, A325763.

Select[Range[100],#==1||EvenQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=FactorInteger[#][[1,2]]&]

A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The a(n) factorizations for n = 36, 360, 720, 192, 288:
  (36)     (360)    (720)     (192)      (288)
  (6*6)    (5*72)   (8*90)    (3*64)     (8*36)
  (2*2*9)  (8*45)   (9*80)    (4*48)     (9*32)
  (3*3*4)  (9*40)   (10*72)   (6*32)     (16*18)
           (10*36)  (16*45)   (12*16)    (2*144)
           (5*8*9)  (5*144)   (3*8*8)    (6*6*8)
                    (5*9*16)  (4*6*8)    (2*2*72)
                    (8*9*10)  (3*4*16)   (2*9*16)
                              (3*4*4*4)  (3*3*32)
                                         (2*2*8*9)
                                         (3*3*4*8)
                                         (2*2*2*36)
                                         (2*2*2*2*2*9)

A001055 counts factorizations.
A118914 is sorted prime signature.
A124010 is prime signature.
A336737 counts factorizations with intersecting signatures.
Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]!={}&]&]],{n,100}]

A109299 Primal codes of canonical finite permutations on positive integers.

Original entry on oeis.org

1, 2, 12, 18, 360, 540, 600, 1350, 1500, 2250, 75600, 105840, 113400, 126000, 158760, 246960, 283500, 294000, 315000, 411600, 472500, 555660, 735000, 864360, 992250, 1296540, 1389150, 1440600, 1653750, 2572500, 3241350, 3601500, 3858750

Offset: 1

Jon Awbrey, Jul 09 2005

nonn

A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
From Rémy Sigrist, Sep 18 2021: (Start)
As usual with lists, the terms of the sequence are given in ascending order.
Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)

			Writing (prime(i))^j as i:j, we have this table:
Primal Codes of Canonical Finite Permutations
        1 = { }
        2 = 1:1
       12 = 1:2 2:1
       18 = 1:1 2:2
      360 = 1:3 2:2 3:1
      540 = 1:2 2:3 3:1
      600 = 1:3 2:1 3:2
     1350 = 1:1 2:3 3:2
     1500 = 1:2 2:1 3:3
     2250 = 1:1 2:2 3:3
    75600 = 1:4 2:3 3:2 4:1
   105840 = 1:4 2:3 3:1 4:2
   113400 = 1:3 2:4 3:2 4:1
   126000 = 1:4 2:2 3:3 4:1
   158760 = 1:3 2:4 3:1 4:2
   246960 = 1:4 2:2 3:1 4:3
   283500 = 1:2 2:4 3:3 4:1
   294000 = 1:4 2:1 3:3 4:2
   315000 = 1:3 2:2 3:4 4:1
   411600 = 1:4 2:1 3:2 4:3
   472500 = 1:2 2:3 3:4 4:1
   555660 = 1:2 2:4 3:1 4:3
   735000 = 1:3 2:1 3:4 4:2
   864360 = 1:3 2:2 3:1 4:4
   992250 = 1:1 2:4 3:3 4:2
  1296540 = 1:2 2:3 3:1 4:4
  1389150 = 1:1 2:4 3:2 4:3
  1440600 = 1:3 2:1 3:2 4:4
  1653750 = 1:1 2:3 3:4 4:2
  2572500 = 1:2 2:1 3:4 4:3
  3241350 = 1:1 2:3 3:2 4:4
  3601500 = 1:2 2:1 3:3 4:4
  3858750 = 1:1 2:2 3:4 4:3
  5402250 = 1:1 2:2 3:3 4:4

Suggested by Franklin T. Adams-Watters

J. Awbrey, Riffs and Rotes
Rémy Sigrist, PARI program for A109299

Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.

PARI
```
\\ See Links section.
```

is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ Rémy Sigrist, Sep 18 2021

Offset changed to 1 and data corrected by Rémy Sigrist, Sep 18 2021

A324855 Lexicographically earliest sequence containing 2 and all squarefree numbers > 2 whose prime indices already belong to the sequence.

Original entry on oeis.org

2, 3, 5, 11, 15, 31, 33, 47, 55, 93, 127, 137, 141, 155, 165, 211, 235, 257, 341, 381, 411, 465, 487, 517, 633, 635, 685, 705, 709, 771, 773, 811, 907, 977, 1023, 1055, 1285, 1297, 1397, 1457, 1461, 1483, 1507, 1551, 1621, 1705, 1905, 2055, 2127, 2293, 2319

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   15: {2,3}
   31: {11}
   33: {2,5}
   47: {15}
   55: {3,5}
   93: {2,11}
  127: {31}
  137: {33}
  141: {2,15}
  155: {3,11}
  165: {2,3,5}
  211: {47}
  235: {3,15}
  257: {55}
  341: {5,11}
  381: {2,31}

Robert Israel, Table of n, a(n) for n = 1..1567
Gus Wiseman, The rooted identity trees whose Matula-Goebel numbers are the first 64 terms.

Cf. A000002, A000720, A001462, A079254, A109298, A112798, A276625, A290822.
Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324854.
Contains A007097 except for 1.

S:= {2}: count:= 1:
for n from 3 by 2 while count < 100 do
  F:= ifactors(n)[2];
  if max(map(t -> t[2],F))=1 and {seq(numtheory:-pi(t[1]),t=F)} subset S then
     S:= S union {n}; count:= count+1;
  fi
od:
sort(convert(S,list)); # Robert Israel, Mar 22 2019

aQ[n_]:=Switch[n,1,False,2,True,?(!SquareFreeQ[#]&),False,,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];
Select[Range[1000],aQ]

cp's OEIS Frontend

A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

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A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

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Mathematica

A324587 Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

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Mathematica

A325761 Heinz numbers of integer partitions whose length is itself a part.

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Mathematica

A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

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Mathematica

A276936 Numbers m with at least one distinct prime factor prime(k) such that prime(k)^k divides, but prime(k)^(k+1) does not divide m.

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A325762 Heinz numbers of integer partitions with no part greater than the number of ones.

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Mathematica

A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

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Mathematica

A109299 Primal codes of canonical finite permutations on positive integers.

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